Date of Award

2013

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Graduate Group

Mathematics

First Advisor

Ted Chinburg

Abstract

The goal of this thesis is to explore the interplay between binary self-dual codes and the \'etale cohomology of arithmetic schemes. Three constructions of binary self-dual codes with arithmetic origins are proposed and compared: Construction $\Q$, Construction G and the Equivariant Construction. In this thesis, we prove that up to equivalence, all binary self-dual codes of length at least $4$ can be obtained in Construction $\Q$. This inspires a purely combinatorial, non-recursive construction of binary self-dual codes, about which some interesting statistical questions are asked. Concrete examples of each of the three constructions are provided. The search for binary self-dual codes also leads to inspections of the cohomology ``ring" structure of the \'etale sheaf $\mu_2$ on an arithmetic scheme where $2$ is invertible. We study this ring structure of an elliptic curve over a $p$-adic local field, using a technique that is developed in the Equivariant Construction.

Included in

Mathematics Commons

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